nLab geometric embedding

Idea

A geometric embedding is the right notion of embedding or inclusion of topoi $F \hookrightarrow E$, i.e. of subtoposes.

Notably the inclusion $Sh(S) \hookrightarrow PSh(S)$ of a category of sheaves into its presheaf topos or more generally the inclusion $Sh_j E \hookrightarrow E$ of sheaves in a topos $E$ into $E$ itself, is a geometric embedding. Actually every geometric embedding is of this form, up to equivalence of topoi.

Another perspective is that a geometric embedding $F \hookrightarrow E$ is the localization of $E$ at the class $W$ or morphisms that the left adjoint $E \to F$ sends to isomorphisms in $F$.

The induced geometric morphism of a topological immersion $X \hookrightarrow Y$ is a geometric embedding. The converse holds if $Y$ is a $T_0$ space. (Example A4.2.12(c) in (Johnstone))

Definition

For $F$ and $E$ two topoi, a geometric morphism

$$F \stackrel{f}{\to} E \;\;\;\; F \stackrel{\stackrel{f_*}{\longrightarrow}}{\underset{f^*}{\longleftarrow}} E$$

is a geometric embedding if the following equivalent conditions are satisfied

• the direct image functor $f_*$ is full and faithful (so that $F$ is a full subcategory of $E$);

• the counit $\epsilon : f^* f_* \to Id_{F}$ of the adjunction $(f^* \dashv f_*)$ is an isomorphism

• there is a Lawvere-Tierney topology on $E$ and an equivalence of categories $e : F \stackrel{\simeq}{\to} Sh_j E$ such that the diagram of geometric morphisms $\array{ F &\stackrel{f_*}{\to}& E \\ & {}_{e}\searrow^\simeq & \uparrow^{i} \\ && Sh_j E}$ commutes up to natural isomorphism $e^* i^* \simeq f^*$

That the first two conditions are equivalent is standard, that the third one is equivalent to the first two is for instance corollary 7 in section VII, 4 of (MacLaneMoerdijk)

Properties

Relation to localization

There is a close relation between geometric embedding and localization: reflective localization.

Let $f : F \hookrightarrow E$ be a geometric embedding and let $W \subset Mor(E)$ be the class of morphisms sent by $f^*$ to isomorphisms in $F$.

Theorem

We have:

• $F$ is equivalent to the localization $E[W]^{-1}$;

• $F$ is equivalent to the full subcategory of $E$ on $W$-local objects.

This fact connects for instance the description of sheafification in terms of geometric embedding $Sh(S) \hookrightarrow PSh(S)$ as described for instance in

• MacLane-Moerdijk, Sheaves in Geometry and Logic

with that in terms of localization at local isomorphisms, as described in

• Kashiwara-Schapira, Categories and Sheaves.

Moreover, this is the basis on which sheafification is generalized to (∞,1)-sheafification in

• Jacob Lurie, Higher Topos Theory .

The following gives a detailed proof of the above assertion.

Write $\eta : Id_E \to f_* f^*$ for the unit of the adjunction.

Since $f_*$ is fully faithful we will identify objects and morphism of $F$ with their images in $E$. To further trim down the notation write $\bar {(-)} := f^*$ for the left adjoint.

Definition

Write $W$ for the class of morphism that are sent to isomorphism under $f^*$,

$$W = (f^*)^{-1}\{g: c\stackrel{\simeq}{\to} d \in Mor(E)\} \,.$$

Proposition

$E$ equipped with the class $W$ is a category with weak equivalences, in that $W$ satisfies 2-out-of-3.

Proof

Follows since isomorphisms satisfy 2-out-of-3.

Proposition

$W$ is a left multiplicative system.

Proof

This follows using the fact that $f^*$ is left exact and hence preserves finite limits.

In more detail:

We have already seen in the previous proposition that

• every isomorphism is in $W$;

• $W$ is closed under composition.

It remains to check the following points:

Given any

$$\array{ && a \\ && \downarrow^w \\ b &\stackrel{h}{\to}& c }$$

with $w \in W$, we have to show that there is

$$\array{ d &\to& a \\ \downarrow^{w'} && \downarrow^w \\ b &\stackrel{h}{\to}& c }$$

with $w' \in W$.

To get this, take this to be the pullback diagram, $w' := h^* w$. Since $f^*$ preserves pullbacks, it follows that

$$\array{ \bar d &\to& \bar a \\ \downarrow^{\bar w'} && \downarrow^{\bar w} \\ \bar b &\stackrel{\bar h}{\to}& \bar c }$$

is a pullback diagram in $F$ with $\bar w' = \bar h^* \bar w$. But by assumption $\bar w$ is an isomorphism. Therefore $\bar w'$ is an isomorphism, therefore $w'$ is in $W$.

Finally for every

$$a \stackrel{\stackrel{r}{\to}}{\stackrel{s}{\to}} b \stackrel{w}{\to} c$$

with $w \in W$ such that the two composites coincide, we need to find

$$d \stackrel{w'}{\to} a \stackrel{\stackrel{r}{\to}}{\stackrel{s}{\to}} b$$

with $w' \in W$ such that the composites again coincide.

To get this, take $w'$ to be the equalizer of the two morphisms. Sending everything with $f^*$ to $F$ we find from

$$\bar a \stackrel{\stackrel{\bar r}{\to}}{\stackrel{\bar s}{\to}} b \stackrel{\bar w}{\to} c$$

that $\bar r = \bar s$, since $\bar w$ is an isomorphism. This implies that $\bar w'$ is the equalizer

$$\bar d \stackrel{\bar w'}{\to} a \stackrel{\stackrel{\bar r}{\to}}{\stackrel{\bar s}{\to}} b$$

of two equal morphism, hence an identity. So $w'$ is in $W$.

Proposition

For every object $a \in E$

• the unit $\eta_a : a \to \bar a$ is in $W$;

• if $a$ is already in $F$ then the unit is already an isomorphism.

Proof

This follows from the triangle identities of the adjoint functors.

$$\array{ & \nearrow &\Downarrow^{\eta}& \searrow^{Id_E} \\ E &\stackrel{\bar{(-)}}{\to}& F &\hookrightarrow& E &\stackrel{\bar{(-)}}{\to}& F \\ &&& \searrow &\Downarrow^{\simeq}& \nearrow_{Id_F} } \;\;\;\; = \;\;\;\; \array{ & \nearrow \searrow^{\bar{(-)}} \\ E &\Downarrow^{Id}& F \\ & \searrow \nearrow_{\bar{(-)}} }$$

and

$$\array{ &&& \nearrow &\Downarrow^{\eta}& \searrow^{Id_E} \\ F &\hookrightarrow& E &\stackrel{\bar{(-)}}{\to}& F &\hookrightarrow& E \\ & \searrow &\Downarrow^{\simeq}& \nearrow_{Id_F} } \;\;\;\; = \;\;\;\; \array{ & \nearrow \searrow \\ F &\Downarrow^{Id}& E \\ & \searrow \nearrow }$$

In components they say that

• for every $a \in E$ we have $(\bar a \stackrel{\bar \eta_a}{\to} \bar{\bar a} \stackrel{\simeq}{\to} \bar a) = Id_{\bar a}$

• for every $a \in F$ we have $(a \stackrel{\eta_a}{\to} \bar a \stackrel{\simeq}{\to} a) = Id_a$

This implies the claim.

Definition

An object $a \in E$ is $W$-local object if for every $g : c \to d$ in $W$ the map

$$g^* : Hom_E(d,a) \stackrel{\simeq}{\to} Hom_E(c,a)$$

obtained by precomposition is an isomorphism.

Proposition

Up to isomorphism, the $W$-local objects are precisely the objects of $F$ in $E$

Proof

First assume that $a \in F$. We need to show that $a$ is $W$-local.

Notice that the existence of the required isomorphism $Hom_F(d,a) \simeq Hom_F(c,a)$ is equivalent to the statement that for every diagram

$$\array{ c &\stackrel{}{\to}& d \\ \downarrow^{h} \\ a }$$

there is a unique extension

$$\array{ c &\stackrel{}{\to}& d \\ \downarrow^{h} & \swarrow \\ a } \,.$$

To see the existence of this extension, hit the original diagram with $f^*$ to get

$$\array{ \bar c &\stackrel{\simeq}{\to}& \bar d \\ \downarrow^{\bar h} \\ \bar a \simeq a } \,.$$

By the assumption that $c \to d$ is in $W$ the morphism $\bar c \to \bar d$ here is an isomorphism. By the assumption that $a$ is already in $F$ we have $\bar a \simeq a$ since the counit is an isomorphism. Therefore this diagram clearly has a unique extension

$$\array{ \bar c &\stackrel{\simeq}{\to}& \bar d \\ \downarrow^{\bar h} & \swarrow_{\exists ! k} \\ \bar a \simeq a } \,.$$

By the hom-isomorphism (using full faithfullness of $f_*$ to work entirely in $E$)

$$Hom_E(\bar d, a) \simeq Hom_E(d,a)$$

this defines a morphism $k : d \to a$. Chasing $k$ through the naturality diagram of the hom-isomorphism

$$\array{ Hom_E(\bar d, \bar a) &\stackrel{\simeq}{\to}& Hom_E(d,\bar a) \\ \downarrow && \downarrow \\ Hom_E(\bar c, \bar a) &\stackrel{\simeq}{\to}& Hom_E(c,\bar a) } \,.$$

shows that $k : d \to a$ does extend the original diagram. Again by the Hom-isomorphism, it is the unique morphism with this property.

So $a \in F$ is $W$-local.

Now for the converse, assume that a given $a$ is $W$-local.

By one of the above propositions we know that the unit $\eta_a : a \to \bar a$ is in $W$, so by the $W$-locality of $a$ it follows that

$$\array{ a &\stackrel{\eta_a}{\to}& \bar a \\ \downarrow^{Id_a} \\ a }$$

has an extension

$$\array{ a &\stackrel{\eta_a}{\to}& \bar a \\ \downarrow^{Id_a} & \swarrow_{\rho_a} \\ a } \,.$$

By the 2-out-of-3 property of $W$ shown in one of the above propositions, (using that $Id_a$, being an isomorphism, is in $W$) it follows that $\rho_a : \bar a \to a$ is in $W$.

Since $\bar a$ is in $F$ and therefore $W$-local by the above, it follows that also

$$\array{ \bar a &\stackrel{\rho_a}{\to}& a \\ \downarrow^{Id_{\bar a}} \\ \bar a }$$

has an extension

$$\array{ \bar a &\stackrel{\rho_a}{\to}& a \\ \downarrow^{Id_{\bar a}} & \swarrow_{\lambda_a} \\ \bar a } \,.$$

So $\eta_a$ has a left inverse $\rho_a$ which itself has a left inverse $\lambda_a$. It follows that $\rho_a$ is also a right inverse to $\eta_a$, since

$$\begin{aligned} \stackrel{\rho_a}{\to} \stackrel{\eta_a}{\to} & = \stackrel{\rho_a}{\to} \stackrel{\eta_a}{\to} \underset{id}{\underbrace{ \stackrel{\rho_a}{\to} \stackrel{\lambda_a}{\to} }} \\ & = \stackrel{\rho_a}{\to} \underset{id}{\underbrace{ \stackrel{\eta_a}{\to} \stackrel{\rho_a}{\to} }} \stackrel{\lambda_a}{\to} \\ &= \stackrel{\rho_a}{\to} \stackrel{\lambda_a}{\to} \\ &= Id \end{aligned} \,.$$

So if $a$ is $W$-local we find that $\eta_a : a \to \bar a$ is an isomorphism, hence that $a$ is isomorphic to an object of $F$.

Corollary

$F$ is equivalent to the full subcategory $E_{W-loc}$ of $E$ on $W$-local objects.

Proof

By standard reasoning (e.g. KS lemma 1.3.11) there is a functor $F \to E_{W-loc}$ and a natural isomorphism

$$\array{ F &&\hookrightarrow&& E \\ & \searrow &\Downarrow^{\simeq}& \nearrow \\ && E_{W-loc} } \,.$$

Since $F \hookrightarrow E$ and $E_{W-loc} \hookrightarrow E$ are full and faithful, so is $F \to E_{W-loc}$. Since by the above it is also essentially surjective, it establishes the equivalence $F \simeq E_{W-loc}$.

Proposition

$F$ is equivalent to the localization $E[W^{-1}]$ of $E$ at $W$.

Proof

By one of the above propositions we know that $W$ is a left multiplicative systems.

This implies that the localization $E[W^{-1}]$ is (equivalent to) the category with the same objects as $E$, and with hom-sets given by

$$Hom_{E[W^{-1}]}(a,b) = \underset{a' \stackrel{p \in W}{\to}a}{colim} Hom_E(a',b) \,.$$

There is an obvious candidate for a functor

$$F \to E[W^{-1}]$$

given on objects by the usual embedding by $f_*$ and on morphism by the map which regards a morphism trivially as a span with left leg the identity

$$(a \to b) \;\; \mapsto \;\; \left( \array{ a &\to& b \\ \downarrow^{Id_a} \\ a } \right) \,.$$

For this to be an equivalence of categories we need to show that this is a essentially surjective and full and faithful functor.

To see essential surjectivity, let $a$ be any object in $E$ and let $\eta_a : a \to \bar a$ be the component of the unit of our adjunction on $a$, as above. By one of the above propositons, $\eta_a$ is in $W$. This means that the span

$$\array{ a &\stackrel{Id_a}{\to}& a \\ \downarrow^{\eta_a} \\ \bar a }$$

represents an element in $Hom_{E[W^{-1}]}(\bar a,a)$, and this element is clearly an isomorphism: the inverse is represented by

$$\array{ a &\stackrel{\eta_a}{\to}& \bar a \\ \downarrow^{Id_a} \\ a } \,.$$

Since every $\bar a$ is in the image of our functor, this shows that it is essentially surjective.

To see fullness and faithfulness, let $a, b\in F$ be any two objects. By one of the above propositions this means in particular that $b$ is a $W$-local object. As discussed above, this means that every span

$$\array{ a' &\to& b \\ \downarrow^w \\ a }$$

with $w \in W$ has a unique extension

$$\array{ a' &\to& b \\ \downarrow^w & \nearrow \\ a } \,.$$

But this implies that in the colimit that defines the hom-set of $E[W^{-1}]$ all these spans are identified with spans whose left leg is the identiy. And these are clearly in bijection with the morphisms in $Hom_E(a,b) \simeq Hom_F(a,b)$ so that indeed

$$Hom_{E[W^{-1}]}(a,b) \simeq Hom_{F}(a,b)$$

for all $a,b \in F$. Hence our functor is also full and faithful and therefore define an equivalence of categories

$$F \stackrel{\simeq}{\to} E[W^{-1}] \,.$$

Factorizations and images

There is a factorization system on the 2-category Topos whose left class is the surjective geometric morphisms and whose right class is the geometric embeddings. The factorization of a geometric morphism can be said to construct its image in the topos-theoretic sense.

See geometric surjection/embedding factorization.

Moreover, each geometric embedding itself has a (dense,closed)-factorization.

Related concepts

In the more general context of (∞,1)-topos theory an $(\infty,1)$-geometric embedding is an (∞,1)-geometric morphism

$$(f^* \dashv f_*) : \mathcal{X} \stackrel{\leftarrow}{\hookrightarrow} \mathcal{Y}$$

such that the right adjoint direct image $f_*$ is a full and faithful (∞,1)-functor.

See reflective sub-(∞,1)-category for more details.

References

Section VII, 4 of

• Saunders Mac Lane and Ieke Moerdijk, Sheaves in Geometry and Logic

and section A4.2 of

• Peter Johnstone, Sketches of an elephant: a topos theory compendium.